Question

[Ngọc Anh Trần Thị]

So sánh A và B, biết A= 10²⁰¹⁰+1/10²⁰¹¹ +1 và B= 10²⁰¹¹+1/10²⁰¹²+1

Answer 1

[Ngô Bảo Châu]

Cho C=\(10^{2010}+\frac{1}{10^{2010}}\)

Xét \(A_1=10^{2010}+\frac{1}{10^{2011}}\)và \(B^{ }_1=10^{2011}+\frac{1}{10^{2012}}\)

Ta có \(A_1-C=10^{2010}+\frac{1}{10^{2010}}-10^{2010}-\frac{1}{10^{2010}}\)

         \(A_1-C=10.\left(\frac{1}{10^{2011}}-\frac{1}{10^{2010}}\right)\)

Giair tượng tự ta được \(B_1-C=10^{2010}.\left(9+\frac{1}{10^{2012}}-\frac{1}{10^{2010}}\right)\)

Ta thấy \(\frac{1}{10^{2012}}-\frac{1}{10^{2010}}<\frac{1}{10^{2011}}-\frac{1}{2010}\)\(\Leftrightarrow\frac{1}{10^{2012}}<\frac{1}{10^{2011}}\Rightarrow9+\frac{1}{10^{2012}}>\frac{1}{10^{2011}}\)

=> A1-C<B1-C=>A1<B1=> A1+1<B1+1 HAY A<B